For convex ERM, we can use [1]} to instantiate the oracle. In this case, accuracy \(\alpha _{\text{private}}\)  is the excepted excess empirical risk, which is \(\alpha _{\text{private}}(n,\epsilon ,\delta ) = O{\frac{GD\sqrt{d}\sqrt{\log {1/\delta }}}{n \epsilon }}\) . Using alg:interpolate, given \(0\le \rho \le 1\) , at any point in the stream, we have,
\({\hat{F}_S(\hat{}_S)- \hat{F}_S(^*_S)} &\le \alpha _\text{private}{\frac{n}{2},\rho \epsilon ,\delta _{\text{group}}{\left\lfloor \frac{1}{\rho }\right\rfloor , \rho \epsilon , \rho \delta }} \\&\le O{\frac{GD \sqrt{d} \sqrt{\log {(1/\rho \delta ) \exp {\rho \epsilon {\left\lfloor \frac{1}{\rho }\right\rfloor -1}}}}}{n \rho \epsilon }} \\& \le O{GD {\frac{\sqrt{d}\sqrt{\log {1/\rho \delta }}}{n \rho \epsilon } + \frac{\sqrt{d}}{ n \rho \sqrt{\epsilon }}}}\\& \le O{\frac{GD\sqrt{d}\sqrt{\log {1/\rho \delta }}}{n \rho \epsilon }}\) 
